Two Member Frame - Fixed/Fixed Top Point Load

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two Member Frame - Fixed/Fixed Top Point Load formulas

\(\large{ e  \;\;=\;\; \frac{h}{L}  }\) 
\(\large{ \beta \;\;=\;\; \frac{I_h}{I_v}  }\) 
\(\large{ R_A  \;\;=\;\;  \frac{P\;x^2}{2\;L^3 \; \left( \beta \; e \;+\; 1 \right) } \; \left[ \beta \; e \; \left( 3 \; L - x \right) + 2 \;\left( 3 \; L - 2 \; x \right) \right] }\) 
\(\large{ R_D  \;\;=\;\;  P - R_A  }\)
\(\large{ H_A = H_D \;\;=\;\;  \frac{3\;P\;x^2}{2\;h\;L^2} \; \left( \frac{L\;-\;x}{ \beta\;e \;+\; 1} \right) }\)
\(\large{ M_A \;\;=\;\;  \frac{P\;x^2}{2\;L^2} \; \left( \frac{L\;-\;x}{ \beta\;e \;+\; 1} \right)   }\)
\(\large{ M_B \;\;=\;\;  \frac{P\;x^2}{L^2} \; \left( \frac{L\;-\;x}{ \beta\;e \;+\; 1} \right)   }\)
\(\large{ M_C \;\;=\;\;  R_B\;x - M_B  }\)
\(\large{ M_D \;\;=\;\;  \frac{P\;x \;\left( L \;-\; x \right)  }{ 2\;L^2 }  \; \left(   \frac{ \beta\;e\;\left( 2\;L \;-\; x \right) \;+\;2\;\left( L \;-\; x \right)   }{\beta \;e\;+\;1}  \right)  }\)

Where:

 Units English Metric
\(\large{ h }\) = height of frame \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction point \(\large{in}\) \(\large{mm}\)
\(\large{ H }\) = horizontal reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ I_h }\) = horizontal member second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ I_v }\) = vertical member second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ A, B, C, D, E }\) = point of intrest on frame - -
\(\large{ L }\) = span length under consideration \(\large{in}\) \(\large{mm}\)
\(\large{ P }\) = total concentrated load \(\large{lbf}\) \(\large{N}\)
\(\large{ R }\) = vertical reaction load at bearing point \(\large{lbf}\) \(\large{N}\)

 

diagrams

  • Bending moment diagram (BMD)  -  Used to determine the bending moment at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
  • Free body diagram (FBD)  -  Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
  • Shear force diagram (SFD)  -  Used to determine the shear force at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
  • Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.

 

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Tags: Frame Support